This paper studies a deterministic growth process for Catalan-Stanley trees, a special class of rooted plane trees linked to Dyck paths with odd-length returns, analyzing their growth dynamics and age distribution.
Contribution
It introduces a growth procedure for Catalan-Stanley trees and provides asymptotic analysis of their age and growth speed.
Findings
01
Asymptotic distribution of tree age for large sizes
02
Growth speed comparison between trees and ancestors
03
Characterization of the growth process dynamics
Abstract
Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond to a class of rooted plane trees, so-called Catalan-Stanley trees. This paper investigates a deterministic growth procedure for these trees by which any Catalan-Stanley tree can be grown from the tree of size one after some number of rounds; a parameter that will be referred to as the age of the tree. Asymptotic analyses are carried out for the age of a random Catalan-Stanley tree of given size as well as for the "speed" of the growth process by comparing the size of a given tree to the size of its ancestors.
Equations108
T(z)=1−T(z)z⟺z+T(z)2=T(z),
T(z)=1−T(z)z⟺z+T(z)2=T(z),
S(z,t)=z+1−t−T2zt.
S(z,t)=z+1−t−T2zt.
S(z,t)=z+1−1−T2tz1−T2t,
S(z,t)=z+1−1−T2tz1−T2t,
α(τ)=r⟺τ∈(ρ−1)r(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture) and τ∈(ρ−1)r−1(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture)
α(τ)=r⟺τ∈(ρ−1)r(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture) and τ∈(ρ−1)r−1(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture)
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TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Advanced Combinatorial Mathematics
Full text
\publicationdetails
2020181113964
Growing and Destroying Catalan–Stanley Trees
Benjamin Hackl\affiliationmark1
Helmut Prodinger\affiliationmark2
00footnotetext: B. Hackl is supported by the Austrian
Science Fund (FWF): P 24644-N26 and by the Karl Popper Kolleg
“Modeling-Simulation-Optimization” funded by the Alpen-Adria-Universität Klagenfurt
and by the Carinthian Economic Promotion Fund (KWF). This paper has been written while
he was a visitor at Stellenbosch University.
00footnotetext: H. Prodinger is supported by an incentive grant of the
National Research Foundation of South Africa.
00footnotetext: Email-addresses:
[email protected],
[email protected]
Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Austria
Department of Mathematical Sciences, Stellenbosch University, South Africa
(2017-9-28; 2018-1-31; 2018-2-14)
Abstract
Stanley lists the class of Dyck paths where all returns to the axis are of odd length as
one of the many objects enumerated by (shifted) Catalan numbers. By the standard
bijection in this context, these special Dyck paths correspond to a class of rooted
plane trees, so-called Catalan–Stanley trees.
This paper investigates a deterministic growth procedure for these trees by which any
Catalan–Stanley tree can be grown from the tree of size one after some number of rounds;
a parameter that will be referred to as the age of the tree. Asymptotic analyses are
carried out for the age of a random Catalan–Stanley tree of given size as well as for
the “speed” of the growth process by comparing the size of a given tree to the size of
its ancestors.
keywords:
Planar trees, tree reductions, asymptotic analysis
1 Introduction
It is well-known that the nth Catalan number Cn=n+11(n2n)
enumerates Dyck paths of length 2n. In [9], Stanley lists a
variety of other combinatorial interpretations of the Catalan numbers, one of them being
the number of Dyck paths from (0,0) to (2n+2,0) such that any maximal sequence of
consecutive (1,−1) steps ending on the x-axis has odd length. At this point it is
interesting to note that there are more subclasses of Dyck paths, also enumerated by
Catalan numbers, that are defined via parity restrictions on the length of the returns to
the x-axis as well (see, e.g., [1]). The height of the class of Dyck paths
with odd-length returns to the origin has already been studied
in [8] with the result that the main term of the height
is equal to the main term of the height of general Dyck paths as investigated in [2].
By the well-known glove bijection, this special class of Dyck paths corresponds to a special
class S of rooted plane trees, where the distance between the rightmost
node in all branches attached to the root and the root is odd. This bijection
is illustrated in Figure 1.
The trees in the combinatorial class S are the central object of study in this
paper.
Definition**.**
Let S be the combinatorial class of all rooted plane trees τ, where the
rightmost leaves in all branches attached to the root of τ have an odd
distance to the root. In particular, itself, i.e., the tree consisting of
just the root belongs to S as well. We call the trees in SCatalan–Stanley trees.
There are some recent approaches (see
[6, 7]) in which classical tree parameters like the register function
for binary trees are analyzed by, in a nutshell, finding a proper way to grow tree
families in a way that the parameter of interest corresponds to the age of the tree
within this (deterministic) growth process.
Following this idea, the aim of this paper is to define a “natural” growth process enabling us to
grow any Catalan–Stanley tree from , and then to analyze the corresponding
tree parameters.
In Section 2 we define such a growth process and analyze some
properties of it. In particular, in Proposition 2.4 we
characterize the family of trees that can be grown by applying a fixed number of
growth iterations to some given tree family. This is then used to derive generating functions
related to the parameters investigated in Sections 3 and 4.
Section 3 contains an analysis of the age of Catalan–Stanley trees, asymptotic
expansions for the expected age among all trees of size n and the corresponding variance
are given in Theorem 1.
Section 4 is devoted to the analysis of how fast trees of given size can
be grown by investigating the size of the rth ancestor tree compared to the size of the
original tree. This is characterized in Theorem 2.
We use the open-source computer mathematics software
SageMath [10] with its included module for computing with asymptotic
expansions [5] in order to carry out the
computationally heavy parts of this paper. The corresponding worksheet can be found at
https://benjamin-hackl.at/publications/catalan-stanley/.
2 Growing Catalan–Stanley Trees
We denote the combinatorial class of rooted plane trees with T,
and the corresponding generating function enumerating these trees with respect
to their size by T(z). For the sake of readability, we omit the argument of
T(z)=T throughout this paper. By means of the symbolic
method [4, Chapter I], the combinatorial
class T satisfies the construction T=\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture×SEQ(T). It translates into the functional equation
[TABLE]
which will be used throughout the paper. Additionally, it is easy to see by solving the
quadratic equation in (1) and choosing the correct branch of the
solution, we have the well-known formula T(z)=21−1−4z.
Proposition 2.1**.**
The generating function of the combinatorial class S of
Catalan–Stanley trees, where t marks all the rightmost nodes in the branches attached
to the root of the tree and z marks all other nodes, is given by
[TABLE]
In particular, there is one Catalan–Stanley tree of size 1 and Cn−2
Catalan–Stanley trees of size n for n≥2.
Proof.
By using the symbolic method [4, Chapter I], the symbolic
representation of S given in Figure 2
(which is based on the decomposition of the rightmost path in each subtree of the root
into a sequence of pairs of rooted plane trees and the final rightmost leaf ■)
translates into the functional equation
In order to enumerate Catalan–Stanley trees with respect to
their size, we consider S(z,z), which simplifies to z(T+1) and thus proves
the statement.
∎
We want to describe how to grow all Catalan–Stanley trees beginning from the tree that has
only one node, .
We consider the tree reduction ρ:S→S that operates on a given Catalan–Stanley tree
τ (or just the root) as follows:
Start from all nodes that are represented by t, i.e. the rightmost
leaves in the branches attached to the root: if the node is a child of the root,
it is simply deleted. Otherwise we delete all subtrees of the grandparent of the
node and mark the resulting leaf, i.e., the former grandparent, with t.
This tree reduction is illustrated in Figure 3. While the
reduction ρ is certainly not injective as there are several trees with the same
reduction τ∈S, it is easy to construct a tree reducing to some given
τ∈S by basically inserting chains of length 2 before all rightmost
leaves in the branches attached to the root. This allows us to think of the operator
ρ−1 mapping a given tree (or some family of trees) to the respective set of
preimages as a tree expansion operator. In this context, we also want to define
the age of a Catalan–Stanley tree.
Definition**.**
Let τ∈S be a Catalan–Stanley tree. Then we define α(τ), the
age of τ, to be the number of expansions required to grow τ from the
tree of size one, . In particular, we want
[TABLE]
for r∈Z≥1, and we set α(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture)=0.
Remark*.*
Naturally, the concept of the age of a tree strongly depends on the underlying
reduction procedure. In particular, for the reduction procedure considered in this
article we have α(τ)=r if and only if the maximum depth of the rightmost
leaves in the branches attached to the root is 2r−1.
Before we delve into the analysis of the age of Catalan–Stanley trees, we need to be able
to translate the tree expansion given by ρ−1 into a suitable form so that we can
actually use it in our analysis. The following proposition shows that ρ−1 can be
expressed in the language of generating functions.
Proposition 2.2**.**
Let F⊆S be a family of Catalan–Stanley trees
with bivariate generating function f(z,t), where t marks rightmost
leaves in the branches attached to the root and z marks all other
nodes. Then the generating function of ρ−1(F), the family of
trees whose reduction is in F, is given by
[TABLE]
Proof.
From a combinatorial point of view it is obvious that the operator Φ has to be
linear, meaning that we can focus on determining all possible expansions of some tree
represented by the monomial zntk, i.e. a tree where the root has k children
(and thus k different rightmost leaves in the branches attached to the root), and n other nodes.
In order to expand such a tree represented by zntk we begin by inserting a chain
of length two before every rightmost leaf in order to ensure that the distance to the
root is still odd. These newly inserted nodes can now be considered to be roots of some
rooted plane trees, meaning that we actually insert two arbitrary rooted plane trees
before every node represented by t. This corresponds to a factor of tkT2k.
In addition to this operation, we are also allowed to add new children to the root,
i.e. we can add sequences of nodes represented by t before or after every child of
the root. As observed above, the root has k children and thus there are k+1
positions where such a sequence can be attached. This corresponds to a
factor of (1−t)−(k+1).
Finally, we observe that nodes that are represented by z are not expanded in any way,
meaning that zn remains as it is.
Putting everything together yields that
[TABLE]
which, by linearity of Φ, proves the statement.
∎
Corollary 2.3**.**
The generating function S(z,t) satisfies the functional equation
[TABLE]
Proof.
This follows immediately from the fact that the reduction operator ρ is surjective,
as discussed above.
∎
Actually, in order to carry out a thorough analysis of this growth process for
Catalan–Stanley trees we need to have more information on the iterated application of the
expansion. In particular, we need a precise characterization of the family of
Catalan–Stanley trees that can be grown from some given tree family by expanding it a
fixed number of times.
Proposition 2.4**.**
Let r∈Z≥0 be fixed and F⊆S be a family of
Catalan–Stanley trees with bivariate generating function f(z,t).
Then the family of trees obtained by expanding the trees in Fr times is
enumerated by the generating function
[TABLE]
Proof.
By linearity, it is sufficient to determine the generating function for the family of
trees obtained by expanding some tree represented by zntk. Consider the closely related multiplicative operator Ψ with
[TABLE]
It is easy to see that we can write the r-fold application of Φ with
the help of Ψ as
[TABLE]
As Ψ is multiplicative, we have
[TABLE]
meaning that we only have to investigate the r-fold application of Ψ to
z and to t.
We immediately see that Ψr(z)=z, as Ψ maps z to z
itself. For Ψr(t), we can prove by induction that the relation
[TABLE]
holds for r≥0. Finally, observe that for j≥1 we have
[TABLE]
and thus
[TABLE]
by iteratively using (5) in the numerator. With our explicit
formula for Ψr(t) from above this yields
From this characterization we immediately obtain the generating functions for all the
classes of objects we will investigate in the following sections.
Corollary 2.5**.**
Let r∈Z≥0. The generating function Fr≤(z,t) enumerating
Catalan–Stanley trees of age less than or equal to r where t marks the rightmost
leaves in the branches attached to the root and z marks all other nodes is given by
[TABLE]
Proof.
As we defined ρ(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture)=\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture we have \leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture∈ρ−1(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture), which implies Fr≤(z,t) is given by Φr(z).
∎
Corollary 2.6**.**
Let r≥0. Then the generating function Gr(z,v) enumerating
Catalan–Stanley trees where z marks the tree size and v marks the size of
the r-fold reduced tree, is given by
[TABLE]
Proof.
Observe that the generating function S(zv,tv) enumerates Catalan–Stanley trees with
respect to the number of rightmost leaves in the branches attached to the root (marked by
t), the number of other nodes (marked by z), and the size of the tree (marked by
v). Applying the operator Φr to this generating function thus yields a
generating function where v still marks the size of the tree, t and z however
enumerate the number of rightmost leaves in the branches attached to the root and all
other nodes of the r-fold expanded tree, respectively. After setting t=z, we obtain
a generating function where v marks the size of the original tree and z the size of
the r-fold expanded tree—which is equivalent to the formulation in the corollary.
∎
3 The Age of Catalan–Stanley Trees
In this section we want to give a proper analysis of the parameter α defined in the
previous section. Formally, we do this by considering the random variable Dn
modeling the age of a tree of size n, where all Catalan–Stanley trees of size n are
equally likely.
Remark*.*
It is noteworthy that in [7] it was shown
that the well-known register function of a binary tree can also be obtained as the
number of times some reduction can be applied to the binary tree until it
degenerates. The age of a Catalan–Stanley tree can thus be seen as a “register
function”-type parameter as well.
First of all, we are interested in the minimum and maximum age a tree of size n can
have.
Proposition 3.1**.**
Let n∈Z≥2. Then the bounds
[TABLE]
hold and are sharp, i.e. there are trees τ, τ′∈S of size n≥2 such that Dn(τ)=1 and Dn(τ′)=⌊n/2⌋ hold. The only tree of size 1 is
, and it satisfies D1(\leavevmodeto7.47pt\vboxto7.47pt\pgfpicture\makeatletter\lower-3.73589ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfont\pgfsys@beginscope\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscopeto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@moveto3.53589pt0.0pt\pgfsys@curveto3.53589pt1.95284pt1.95284pt3.53589pt0.0pt3.53589pt\pgfsys@curveto-1.95284pt3.53589pt-3.53589pt1.95284pt-3.53589pt0.0pt\pgfsys@curveto-3.53589pt-1.95284pt-1.95284pt-3.53589pt0.0pt-3.53589pt\pgfsys@curveto1.95284pt-3.53589pt3.53589pt-1.95284pt3.53589pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@stroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.00.0pt0.0pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture)=0.
Proof.
Note that only , the tree of size 1 has age [math], therefore the lower bound
is certainly valid for trees of size n≥2. This lower bound is sharp, as the tree
with n−1 children attached to the root is a Catalan–Stanley tree and has age 1.
For the upper bound, first observe that given a tree of size n≥3 the reduction
ρ always removes at least 2 nodes from the tree. If the tree is of size 2, then
ρ only removes one node. Given an arbitrary Catalan–Stanley tree τ of age r
and size n, this means that
[TABLE]
where ∣τ∣ denotes the size of the tree τ. This yields r≤n/2,
and as r is known to be an integer we may take the floor of the number on the
right-hand side of the inequality. This proves that the upper bound
in (8) is valid.
The upper bound is sharp because we can construct appropriate families of trees precisely
reaching the upper bound: for even n, the chain of size n is a Catalan–Stanley tree
of age n/2. For odd n=2m+1 we consider the chain of size 2m and attach one
node to the root of it. The resulting tree is a Catalan–Stanley tree of age m=⌊n/2⌋, and thus proves that the bound is sharp.
∎
By investigating the generating functions obtained from Corollary 2.5 we
can characterize the limiting distribution of the age of Catalan–Stanley trees when the
size n tends to ∞.
Theorem 1**.**
Consider n→∞. Then the age of a (uniformly random) Catalan–Stanley tree of size
n behaves according to a discrete limiting distribution where
[TABLE]
for r∈Z≥1, and the O-term holds uniformly in r. Additionally, by setting
[TABLE]
the expected age and the corresponding variance are given by the asymptotic expansions
[TABLE]
[TABLE]
Proof.
For the sake of convenience we set Fr≤(z):=Fr≤(z,z), where
Fr≤(z,t) is given in (6). This univariate generating
function now enumerates Catalan–Stanley trees of age ≤r with respect to the tree
size.
We begin by observing that Fr≥(z), the generating function enumerating
Catalan–Stanley trees of age ≥r with respect to the tree size is given by
[TABLE]
where the last equation follows after some elementary manipulations and by using (1).
Now let fn,r:=[zn]Fr≥(z) denote the number of Catalan–Stanley trees
of size n and age ≥r. As we consider all Catalan–Stanley trees of size n to be
equally likely, we find
[TABLE]
We use singularity analysis (see [3]
and [4, Chapter VI]) in order to obtain an asymptotic
expansion for fn,r. To do so, we first observe that z=1/4 is the dominant
singularity of T and thus also of Fr≥(z). We then
consider z to be in some Δ-domain at
1/4 (see [4, Definition VI.1]). The task of expanding
Fr≥(z) for z→1/4 now largely consists of handling the term
1+T2r−1T2r−1. Observe that we can write
[TABLE]
which results in
[TABLE]
where the O-term holds uniformly in r. Multiplying this expansion with the expansion
of z(1+T) yields the expansion
[TABLE]
By means of singularity analysis we extract the nth coefficient and find
[TABLE]
Computing the difference fn,r−fn,r+1 and dividing by the Catalan number
Cn−2 then yields the expression for P(Dn=r) given in (9).
The expected value can then be computed with the help of the well-known formula
[TABLE]
which proves (10). Finally, the variance can be obtained from VDn=E(Dn2)−(EDn)2, where
In addition to the asymptotic expansions given in Theorem 1 we can
also determine an exact formula for the expected value EDn. The key tools in this
context are Cauchy’s integral formula as well as the substitution z=(1+u)2u.
Proposition 3.2**.**
Let n∈Z≥2. The expected age of the Catalan–Stanley trees of size n is
given by
[TABLE]
where σ0odd(k) denotes the number of odd divisors of k.
Proof.
We begin by explicitly extracting the coefficient [zn]Fr≥(z). The
expected value can then be obtained by summation over r and division by Cn−2.
With the help of the substitution z=(1+u)2u we can bring
Fr≥(z) into the more suitable form
[TABLE]
We extract the coefficient of zn now by means of Cauchy’s integral formula. Let
γ be a small contour winding around the origin once. Then we have
[TABLE]
where γ~, the integration contour of the second integral, is the
transformation of γ under the transformation z=u/(1+u)2 and is also a
small contour winding around the origin once.
Now consider the auxiliary sum
[TABLE]
It is easy to see by distinguishing between even and odd k that with the help of
σ0odd(k),
ϑ(k) can be written as ϑ(k)=(−1)k−1σ0odd(k).
Summing the expression from (16) over r≥1, simplifying the
resulting double sum by means of the auxiliary sum ϑ, and finally dividing by
Cn−2 then proves (13).
∎
4 Analysis of Ancestors
In this section we focus on characterizing the effect of the (repeatedly applied)
reduction ρ on a random Catalan–Stanley tree of size n. We are particularly
interested in studying the size of the reduced tree. In the light of the fact that all
Catalan–Stanley trees can be grown from by means of the growth process
induced by ρ, we can think of the rth reduction of some tree τ as the rth
ancestor of τ.
In order to formally conduct this analysis, we consider the random variable Xn,r
modeling the size of the rth ancestor of some tree of size n, where all Catalan–Stanley
trees of size n are equally likely.
Similar to our approach in Proposition 3.1 we can determine precise
bounds for Xn,r as well.
Proposition 4.1**.**
Let n∈Z≥2 and r∈Z≥1. Then the bounds
[TABLE]
hold for r≤⌊n/2⌋ and are sharp, i.e. there are trees τ,
τ′∈S of size n≥2 such that Xn,r(τ)=1 and
Xn,r(τ′)=n−2(r−1)−1. For r>⌊n/2⌋ the variable Xn,r is
deterministic with Xn,r=1.
Proof.
Assume that r≤⌊n/2⌋. The lower bound is obvious as trees cannot
reduce further than to , and as the first ancestor of the tree with n−1
children attached to the root already is the lower bound is valid and
sharp.
For the upper bound we follow the same argumentation as in the proof of
Proposition 3.1 to arrive at
[TABLE]
for some Catalan–Stanley tree of size n, which proves that the upper bound is
valid. Any tree τ of size n having the chain of length 2 as its (r−1)th ancestor
satisfies Xn,r(τ)=n−2(r−1)−1 and thus proves that the upper bound is
sharp. This proves (17).
In the case of r>⌊n/2⌋ we observe that as the ⌊n/2⌋th
ancestor of any Catalan–Stanley tree of size n already is certain to be
by Proposition 3.1, the rth ancestor is as well.
∎
With the generating function Gr(z,v) enumerating Catalan–Stanley trees with respect
to their size (marked by n) and the size of their rth ancestor (marked by v) from
Corollary 2.6 we can write the probability generating function of
Xn,r as
[TABLE]
This allows us to extract parameters like the expected size of the rth ancestor and the
corresponding variance.
Theorem 2**.**
Let r∈Z≥0 be fixed and consider n→∞. Then the expected value and
the variance of the random variable Xn,r modeling the size of the rth ancestor
of a (uniformly random) Catalan–Stanley tree of size n are given by the asymptotic
expansions
[TABLE]
[TABLE]
Proof.
The strategy behind this proof is to determine the first and second factorial moment of
Xn,r by extracting the coefficient of zn in the derivatives
∂vd∂dGr(z,v)∣v=1 for d∈{1,2} and normalizing the
result by dividing by Cn−2.
We begin with the expected value. With the help of SageMath [10] we
find for z→1/4
[TABLE]
where the O-constant depends implicitly on r. Extracting the coefficient of zn
and dividing by Cn−2 yields the expansion given
in (18).
Following the same approach for the second derivative yields the expansion
[TABLE]
such that after applying singularity analysis and division by Cn−2 we obtain the
expansion
[TABLE]
for the second factorial moment EXn,r2. Applying the well-known
formula
[TABLE]
then leads to the asymptotic expansion for the variance given
in (19) and thus proves the statement.
∎
Besides the asymptotic expansion given in Theorem 2, we are also
interested in finding an exact formula for the expected value EXn,r. We can do so
by means of Cauchy’s integral formula.
Proposition 4.2**.**
Let n, r∈Z≥1. Then the expected size of the rth ancestor of a random
Catalan–Stanley tree of size n is given by
[TABLE]
Proof.
We rewrite the derivative g(z):=∂v∂Gr(z,v)∣v=1 into a more suitable
form which makes it easier to extract the coefficients. To do so, we use the
substitution z=u/(1+u)2 again, allowing us to express the derivative as
[TABLE]
Note that as T=1+uu, the summand (1+u)3(1+2u)u actually
represents z(1+T), implying that the coefficient of zn in this
summand is given by Cn−2.
Now let γ be a small contour winding around the origin once, so that with
Cauchy’s integral formula we obtain
[TABLE]
where γ~ is the image of γ under the transformation (and is still a
small contour winding around the origin once). Dividing by Cn−2 then proves (20).
∎
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