If our neighbor’s kids keep coming into our yard without permission and climbing on the trees, how can I protect myself from liabilty of injury?

Generally speaking, a homeowner is not liable for injuries by unauthorized trespassers unless the homeowner caused or failed  to correct a dangerous condition. However, with children, there is a doctrine called "attractive nuisance," which holds that homeowners can be held liable for condiitions potentially dangerous to children which are also attractive to children. The main example of this are swimming pools; I'm not personally familiar with any cases of this doctrine being applied to climbing trees, but it's not impossible that if a child is injured climbing on a tree when you know children like to come and climb it, that the children's parents may at least try to hold you  liable for sue you.

Some good general protective steps to take include:

1) If possible, trim off the lower branches to make climbing impossible or less inviting; consider cutting down trees which are particular issues.

2) Put up a fence with a locking gate around your yard.

3) Make it clear to neighborhood children and parents that they are not welcome to come and climb--e.g. "no trespassing" signs; send an email or letter to parents saying that your trees are not for climbing.

The goal of the above is to show you are taking reasonable steps to keep kids out; all you are expected to do, even if something is deemed an attractive nuisance, is to take reasonable steps.

4) Have a good homowner's policy.