I used to climb the trees when a child... can be understood, but it is better to say ... I used to climb trees when I was a child. Your version is the full version and the good one. The first version is just shortened version of yours. It is sloppy English.

The version you suggest is more common, certainly, and is correct. The first version "when a child" is far less common, and perhaps slightly poetic. One point: only use "the" if you are talking about some specific trees. If it's a more general thing, you'd be better to say "I used to climb trees...".