Definition of infinite tree in set theory

Question

Really basic question concerning trees in set theory.

What is the definition of an infinite tree?

I ask the following because, rather peculiarly, neither in Kechris classical book on descriptive set theory, nor in Srivastava book on the same topic, there is an explicit definition, even if they both use the concept (e.g. Srivastava use the infinity of a tree for the Koenig's infinity lemma, p.28).

Thanks in advance.

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Edit after the answers:
Thanks all for the number of answers, and the quality behind them.

Answer 1

Recall that a tree is a partial order that for each point $x$, the cone below $x$ is well-ordered.

An infinite tree is a tree with infinitely many nodes.

Sometimes, however it is easier to work with "reversed" trees. With reversed ordering, being well-founded is equivalent to not having an infinite branch. And that is something which comes up often in descriptive set theory.

Answer 2

Typically, in set theory language, a (possibly infinite) "tree" means a nonempty set of sequences of symbols (for example integers), which is closed under taking prefixes.

What does this have to do with the graph-theoretic concept of a tree? Assume that you're labeling all edges in the tree with a symbol such that no node has two outgoing edges with the same label. The set theorists' representation of the tree then consists of an "address" of each node in the tree, where the address is the sequence of edge labels on the (unique) path from the root to the node in question.

The root node is addressed by the empty sequence.

The complete binary tree with 7 nodes would be $$ \{(), (0), (0,0), (0,1), (1), (1,0), (1,1) \} $$ and an infinite unary tree would be $$ \{(), (0), (0,0), (0,0,0), \ldots \} $$

The tree is infinite if it has inifinitely many nodes.

Additional constraints on the tree can be defined, such as each node must have finitely many successors, or that there's a global finite limit on the number of successors to a node.

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As Asaf points out, it is sometimes useful to allow transfinitely long (but still well-ordered) sequences as addresses. This gives rise to trees that don't correspond directly to graph-theoretic trees (because nodes whose address is indexed by a limit ordinal won't have immediate parents).

Answer 3

As asked, you actually need to specify a little bit more to be precise. I'll first assume you're asking about infinite, binary branching trees. A binary branching tree $T$ is a collection of finite strings of zeros and ones, called nodes, such that, whenever a string $\sigma$ is in $T$, all of its prefixes are in $T$. An infinite binary branching tree is a binary branching tree with infinitely many strings in it (each individual string is finite).

More generally, you can talk about trees branching on an arbitrary set $A$. Then, rather than considering (sets of) strings of zeros and ones, you consider (sets of) strings of elements of $A$. The definition of an infinite tree is the same: a tree containing infinitely many strings.