# The Combinatorics of Directed Planar Trees

## TL;DR

This paper provides a geometric realization of polyhedra related to directed planar trees, including associahedra, demonstrating their topological properties and calculating key combinatorial invariants like generalized Catalan numbers.

## Contribution

It introduces an explicit geometric realization of polyhedra governed by directed planar trees, extending known structures like associahedra and analyzing their topological and combinatorial properties.

## Key findings

- Polyhedra are homeomorphic to balls.
- Explicit formulas for the number of vertices of generalized associahedra.
- Connections to Catalan numbers and algebraic structures.

## Abstract

We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are homeomorphic to balls. We also calculate the number of vertices of the lowest generalized associahedra, giving appropriate generalizations of the Catalan numbers.