The coding of compact real trees by real valued functions

TL;DR

This paper characterizes how compact real trees with additional structures can be uniquely encoded by specific real-valued functions, extending probabilistic models and analyzing their properties through this coding.

Contribution

It proves that any compact real tree with a root, linear order, and measure can be uniquely encoded by a specific class of real-valued functions, generalizing previous contour process representations.

Findings

Any compatible compact real tree can be encoded by a left-continuous function without positive jumps.

The encoding is unique under minimal backtracking during exploration.

Measure changes on the tree correspond to re-parametrizations of the coding function.

Abstract

This paper is a detailled study of the coding of real trees by real valued functions that is motivated by probabilistic problems related to continuum random trees. Indeed it is known since the works of Aldous (1993) and Le Gall (1991) that a continuous non-negative function $h$ on $[0,1]$ such that $h(0)=0$ can be seen as the contour process of a compact real tree. This particular coding of a compact real tree provides additional structures, namely a root that is the vertex corresponding to $0\in [0,1]$, a linear order inherited from the usual order on $[0,1]$ and a measure induced by the Lebesgue measure on $[0,1]$; of course, the root, the linear order and the measure obtained by such a coding have to satisfy some compatibility conditions. In this paper, we prove that any compact real tree equipped with a root, a linear order and a measure that are compatible can be encoded by a non-negative function $h$ defined on a finite interval $[0, M]$, that is assumed to be left-continuous with right-limit, without positive jump and such that $h(0+)=h(0)=0$. Moreover, this function is unique if we assume that the exploration of the tree induced by such a coding backtracks as less as possible. We also prove that a measure-change on the tree corresponds to a re-parametrization of the coding function. In addition, we describe several path-properties of the coding function in terms of the metric properties of the real tree.