Edit 2: Confirmation that it's one tree!
Surrounding trees on either side along the sidewalk are completely different, and the ones on the other side of the fence are short and have rounded but far smaller, as well as thicker, dark leaves.
I've taken a 180 degree vertical "panoramic" photograph to show that there are no encroachments from trees with large leaves from behind. I've also included a few more photos of small clusters of leaves coming from a single large branch, again showing all three types of leaves.
Note some of the photos are large, and if you open them in a separate tab or window you can zoom in for additional detail.
I was riding a bike in Taipei recently (November 2017, location in google maps - more views of trees: https://goo.gl/oXPHr3) and I saw two huge leaves on the sidewalk. I stopped to photograph them and realized that they were very different shapes but otherwise looked quite similar. They are both over 40 cm long broad leaves, but while one has smooth edges and is "leaf-shaped", the other has long serrations (roughly 10 cm deep, 5-7cm spacing) on each edge.
I took some more photos of the leaves and branches of this very large tree, and one of the trunk, which looks quite gnarly.
Edit 1: I've cropped and lightened sections of two of the photos where I believe I can show examples where leaves on the same branch exhibit continuous change from smoothed edges to serrated edges. I've used
1 to denote smooth-edged leaves,
2 to denote leaves that show a few "digits" at the far end of the leaf, and
3 to denote leaves that exhibit full "digitization". In each case I think it's pretty clear that they connect to the same branch, and are not from different trees as suggested in the answer. This is strongly (if superficially) reminiscent of the sassafras leave differences.
Question: What kind of tree is this, and why do its huge leaves change shape and "grow fingers"?
It was pointed out to me that Sassafras leaves also come in different shapes with "fingers", I've included one image of those below as well for comparison.