Many thanks in advance!

The first time I saw this cactus was in the mid 1980's, it's much older than that. It was pretty much this height then, it's about 8-10' tall.

The garden was replanted in the early 1960's, but the house and parts of the garden are from 1910. So I have no idea how old it is.

8 years ago several outer trunks collapsed, and were left in place, you can see now many babies have sprouted from them.
It's in partial shade, dappled sunlight under an old oak tree. It only gets supplemental water about once a month in the summer, none the rest of the year. Northern California, near San Jose.

Anyone have an idea of what it is? Someone told me Pipe cactus, is this right? Thanks again!