Tree (descriptive set theory)

In descriptive set theory, a tree on a set $X$ is a subset $T$ of $X^{<\omega }$ (that is, a set of finite sequences of elements of $X$) that is closed under subsequence (that is, if $<x_{0},x_{1},\ldots ,x_{n-1}>\in T$ and $m<n$, then $<x_{0},x_{1},\ldots ,x_{m-1}>\in T$). A branch through $T$ is an infinite sequence ${\vec {x}}\in X^{\omega }$ of elements of $X$ such that, for every natural number $n$, ${\vec {x}}|n\in T$, where ${\vec {x}}|n$ denotes the sequence of the first $n$ elements of ${\vec {x}}$.