= Solution
{c}
A finite <tree> is a nonempty <connected graph> with no <graph cycles>. We show by <mathematical induction> on its number $v$ of <vertices> that it has $v-1$ <edges>. For $v=1$ there are no <edges>, so the assertion holds.
If $v>1$, take a <path in a graph> of maximal length among those without repeated <vertices>. Each endpoint is a <leaf of a tree>: an additional incident <edge> would either extend the <path in a graph> or create a <graph cycle>. Delete one such <leaf of a tree> and its unique incident <edge>. The remaining <graph> stays <connected>, since a <path in a graph> between remaining <vertices> never needs that terminal <edge>, and it has no <graph cycles>. It is therefore a <tree> with $v-1$ <vertices> and, by <mathematical induction>, $v-2$ <edges>. Restoring the deleted <edge> gives $e=v-1$.
Thus its <Euler characteristic> is
$$
\boxed{\chi(X)=v-e=v-(v-1)=1.}
$$
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