# Tree-decorated planar maps

## TL;DR

This paper introduces tree-decorated planar maps, establishes a bijection with trees and maps with simple boundaries, and derives explicit counting formulas for various decorated maps.

## Contribution

It presents a novel bijection between tree-decorated planar maps and a product of trees and simple boundary maps, enabling enumeration of these structures.

## Key findings

- Counted tree-decorated triangulations and quadrangulations with fixed faces and tree size.
- Established bijections to facilitate enumeration of decorated planar maps.
- Derived explicit formulas for counting various decorated maps.

## Abstract

We introduce the set of (non-spanning) tree-decorated planar maps, and show that they are in bijection with the Cartesian product between the set of trees and the set of maps with a simple boundary. As a consequence, we count the number of tree decorated triangulations and quadrangulations with a given amount of faces and for a given size of the tree. Finally, we generalise the bijection to study other types of decorated planar maps and obtain explicit counting formulas for them.