= Solution
Here each $T_i$ is a nonempty <connected subgraph of a tree>, rather than a descendant-closed data-structure <subtree>. We prove the <Helly property for connected subgraphs of a tree> by <mathematical induction> on the number of <vertices> of $T$.
For a one-<vertex> <tree>, every nonempty $T_i$ contains that <vertex>. For a larger <tree>, choose a <leaf of a tree> $v$ using part (i), and denote its unique neighbour by $w$. If some $T_i$ consists only of $v$, the assumed pairwise intersections force every $T_j$ to contain $v$, and we are done.
Otherwise, remove $v$ from every $T_i$ that contains it, leaving all other $T_i$ unchanged. Any affected $T_i$ has at least two <vertices>; connectedness forces it to contain $w$ and the <edge> $vw$. Thus $v$ is a <leaf of a tree> of $T_i$, and part (ii) shows that its deletion leaves a nonempty <tree>. All reduced $T_i$ are therefore nonempty <connected subgraphs of a tree> $T\setminus v$.
They still intersect pairwise. An original common <vertex> other than $v$ survives. If a pair had only $v$ in common, both members would also contain $w$, by the previous paragraph, contradicting that it was their only common <vertex>. The induction hypothesis now supplies a common <vertex> of the reduced family, which is also common to the original family. Consequently
$$
\boxed{\bigcap_{i=1}^kV(T_i)\ne\varnothing.}
$$
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