A Cat Named Tree is book two in The Little Netherton Book series by Mary Ann Netherton.  This adorable series is much like the little book series you had as a child.  

In book two, Tree the cat likes to play, but more than anything, he likes climbing.  When he was three years old, he got stuck in a big tree. That is how he got his name

Netherton, Mary Ann - Hardbook - A Cat Named Tree