Up the spout is the only idea I've heard but for me that means being pregnant and the clue suggests something subsequent; also spout doesn't seem to have any relation to wood or trees.
Rusty, my Revised 13th Edition has 'up the stick' at the very end of the definition for 'stick 1' (I think that's all the noun stuff, Stick 2 seems to be the verb lot.) It follows 'up sticks'.