I recently moved and a tree that was blown over in a storm has been growing again for a few years. This is my first summer with the tree and it looks great, until you see the trunk. I think I will have to remove it, but wanted to ask more informed people. Is there any way to save this tree? Or should I remove it as soon as possible?
From the leaf shape, the sprouting at the base and its overall pyramidal shape it looks like a Little Leaf Linden or Tilia cordata variety. Tilia is known for its ability to survive heavy pruning and to largely recover from accidents such as these. It looks like someone has already attempted to prune out some large branches, likely designed to rebalance as much as possible the upper growth to make it more stable and less likely to break again. From the damage left it appears it had two competing leaders, and one collapsed leaving the competitor in place. This probably explains the reasonably new fence. It was likely a large tree about 30-40 feet, and would have been taller with a single leader, headed to a hundred feet. Such a tree will have largely exhausted resources in that corner of the yard, so chances are any further growth will be restrained. Current growth is dense and if it survives a few more years while the trunk strengthens from its current weakened state could go on for many years, but will always be an item of curiosity and carry with it a history that you might have to repeat many times. Inevitably there will be a constant need to keep clearing up the basal shoots at least for the next few years.
If you can easily afford to do so, take it out, including the root, and choose a replacement. If not, I think it will last a while until you can afford to replace.
As it gets bigger it will increasingly be an accident waiting to happen, especially as it could fall over your boundary fence into someone else's property.
Whether it is "worth saving" depends on your personal attitude. If you want "a nice looking tree", then get rid of it and plant a new one. On the other hand, any garden can have "nice looking trees," but you have got something unique.
If you want to make some use of it, you could cut it down at the point where the new growth starts at the top of the trunk, and use the trunk as a support for some climbing plants. You should get a few years use out of it that way, and it won't be supporting any heavy loads so even if it does rot through and collapse eventually it won't do much damage.
The is obviously enough intact trunk to support some strong regrowth, so it you cut it down to the top of the trunk it will continue to throw out new branches from that point, since the root system is presumably undamaged. You may also get some growth from lower down, like the small amount you already have, but it will take decades for that to grow into something that looks like "a tree", and it may never do that.