Here's a link to a picture of the same bike taken several years earlier.
Actually more like pushing up.
(This paragragh is written after everything above and below. Sorry for the long post, but this has interested me since I was a little kid.)
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If something's sitting on a branch or a root and it's not too heavy, yes, since the growth layer stays below it. But once the tree grows around it and has a grip on it, the growth layer is above it and there will be little or no moving.
If that bike was placed in a crotch, it could be pushed up a bit again since the growth layer is below it but once it is surrounded, it's not going anywhere.
Again, it the other pics where things are raised a few inches, it is because the tops of the roots are pushing the object up.
The park bench and the tub are now trapped at the top and bottom and now they aren't going anywhere.
Some guy on the original link said he carved his initials years ago and now they are 15 feet up. No way. If you put a scratch on a tree 3 feet up or drive a spike in it, years later, it will be at about the same level, assuming the ground level doesn't change. I imagine trees push themselves up out of the ground a bit since there's less resistance, but I don't think it's usually going to be much.
On the property we recently bought, the previous owner was not good with fences and trees. In numerous places, chain link leaned against trees and some are now deeply encased. They appear to be at the same height they would have been 10-15 years ago since the tree would only push it sideways a bit until it was encased.
We had a small tree that forked right at ground level under a laid down chain link fence. The two trunks grew up through it but since they were mostly pushing sideways, the chain link was still right on the ground.
The tree was an elm and died. I've still got the trunks. I need to take a pic and post it. It's quite a mess.