Caterpillar decomposition of trees
It seems this is false in general. Take the tree with a central node $x$, connected to a bunch of nodes $y_1,\dots,y_n$. Each node $y_i$ is additionally connected to a node $z_i$.

The total number of nodes is $N=2n+1$.

Any caterpillar in this tree can contain at most two $z$ nodes, and therefore, $n/2$ caterpillars are needed to cover it. So it is impossible to cover it with $O(log N)$ caterpillars.