A functional limit theorem for the profile of $b$-ary trees

TL;DR

This paper establishes a functional limit theorem for the weighted profile of b-ary trees using martingale techniques, connecting continuous and discrete tree profiles.

Contribution

It introduces a new functional limit theorem for b-ary trees' profiles, employing martingales and embedding methods to unify discrete and continuous tree profiles.

Findings

Proves a functional limit theorem for weighted profiles of b-ary trees.

Uses martingale methods linked to branching Markov processes.

Recovers profiles of well-known discrete trees through embedding techniques.

Abstract

In this paper we prove a functional limit theorem for the weighted profile of a $b$-ary tree. For the proof we use classical martingales connected to branching Markov processes and a generalized version of the profile-polynomial martingale. By embedding, choosing weights and a branch factor in a right way, we finally rediscover the profiles of some well-known discrete time trees.