König's Lemma in set theory, why is the finite branching needed? The answers here are quite mathematical. I hope somebody can explain this particular point here.

Why is it for a tree with height $\omega$ that all its levels need to be finite in order to have an infinite branch ? This question is about the necessity of the finiteness of the levels. For example what is false about the picture below ? I don't see why existance or the absence of the (encircled) subset on the right, affects the existance of the infinite branch ?

Definitions
Let $(T,<)$ be a tree, that is $T$ is a partially ordered set such that $\forall t \in T:\ t_< := \{x\in T|x<y\}\ \text{is well-ordered}$.
The height of an element $t$ is the ordinal $ht(t):=\alpha_t \cong t_<$, i.e. of the order-type of $t_<$.
The $\alpha$th level of the tree is $T(\alpha) = \{t\in T|\ ht(t) \cong \alpha\}$
A branch is a subset of $T$ that is maximal chain
The height of the tree is $ht(T) = \sup\{ht(t)+1|\ t \in T\}$,

Added:
I doubt about this last definition, for if we have an $\omega$-long branch, the tree would be $(\omega+1)$-heigh. Though this definition is found also in Kunen, Set Theory, An Introduction to Independence Proofs (1992), §5 Trees, p. 68. Could anyone explain what is wrong with this ?

A tree with a single root and infinitely many children (which all are leaves) is an infinite tree without infinite branch.
