volunteer tree in yard -- SF Bay Area

We think a squirrel probably planted this young tree, although we cannot find one like it nearby. It just started its spring blooming, and has what looks like green caterpillars growing straight up out of its branches. Before this tree lost its leaves for winter, I thought they looked like bay leaves. Does anyone have an idea of what this tree is? Thank you, Lois

 

They look like willow catkins, perhaps the Bay Willow, Salix pentandra?

 

Definitely a willow, anyway. These are dispersed by wind.

 

Agree with a willow; not Bay Willow though, as that is one of the willows that doesn't flower until after leaf-out.

 

So it's just a Bay Area willow.

 

Thanks for your replies! I found a couple of pictures of the tree from last summer. Maybe these will help in clarifying the tree's identification?

 

Still a willow, maybe one of the native species.