Hi!
I got this olive tree from a friend who moved (and wasn't a great gardener). It died from frost a few years ago but has since sprouted some new branches from the trunk up untill where the dead branches split. Question is, what is best to do? Cut back the dead branches of leave them be? I would eventually like it to be a tree again and not an olive bush.
Thanks in advance!

I would cut off each dead branch 6" above the newer green growth. You would immediately remove them from your view just leaving green growth. A few of these new shoots may become dominant and grow upward.
Using fertilizer spikes that can be inserted into the soil at the base of the olive might help.

It can be a tree again.👍👍😀
First. I'd get rid of all that bottom growth.
Then :
Cut all the dead branches off, SHORT ! About 1/2 inch to green. The less dead wood, the better.

Then work on green branches.
At each cluster of new growth. Trim, all but 3-5 branches off, going in direction you desire the tree to look.
Your sculpting the main scaffolding of tree, by forcing it to grow the way you want it. So keep any undesirable side branches, or undesirable buds cut or broke off. Little buds can be snapped off by hand.

When your small clusters of 3-5 branches get to be 8-10 inches, you can decide, which one or ones you want leave.
When branches reach the desired length you want them to branch off at, simply pinch the tip off.

You get the basic idea.
Carry on buddy. 👍👍
😎😎😎

Thanks for the replies!

wil get some fertilizer to give the tree a boost.
Cut away all of the lower branches and most of the upper as i thought they were ugly.

Hope to get a healty growing tree!