Now I am studying about basic concepts of trees on Descriptive Set Theory.
On Classical Descriptive Set Theory by Alexander Kechris, the definition of tree is written like this:
A tree on a set $A$ is a subset $T\subseteq A^{<\mathbb{N}}$ closed under initial segment; i.e., if $t\in T$ and $s\subseteq t$, then $s\in T$.(In particular, $\emptyset \in T$ if $T$ is nonempty).
Note:
$A^{\mathbb{N}}=\bigcup A^n$ with $n\in \mathbb{N}$ is a set of all finite sequences from $A$.
And $A^n$ is the set of finite sequences $s=(s(0),...,s(n-1))=(s_0,...,s_{n-1})$ of length $n$ from $A$.
By that definition, i still can't imagine how the form of the tree is. Can anyone give me an example of tree? Any help will be so appreciate. Thanks.
Perhaps a better way to phrase it might be that a tree on a set $X$ (in this context - the term "tree" is used differently in others) is a set of finite strings from $X$ which is closed under taking prefixes: e.g. if $X=\mathbb{N}$ and $T$ is a tree on $X$ with $\langle 1,42, 17\rangle\in T$, then we must also have $\langle 1,42\rangle\in T$. Often trees are defined negatively, as the set of strings which don't have some "badness" property - where this property is such that once a string is "bad," making it longer won't make it "less bad." For example, the set of all strings not consisting only of prime numbers is a tree.