Walkthrough

Tree: $1-2$, $2-3$, $3-4$, $4-5$. Target length: $k=2$. Centroid is $3$. Distances from $3$: node $1$ is $2$, node $2$ is $1$, node $3$ is $0$, node $4$ is $1$, node $5$ is $2$. Pairs with sum $2$: node $2$ (distance $1$) with node $4$ (distance $1$). That is one path: $2-3-4$.

Check child subtrees: subtree rooted at $2$ contains $\{1,2\}$, subtree at $4$ contains $\{4,5\}$. No pairs within a single subtree sum to $2$. Recurse on components: each is small, no additional paths. Total: $1$ path of length $2$.