Trees have branch lengths and a pattern of connectivity, referred to as topology. In addition trees can be rooted or unrooted. Often trees are depicted as rooted, but should be considered as unrooted.

The following five rooted trees all have a different tree topology, because the root is placed in a different location, and in addition the trees are different, because the branch lenths are different.

BUT if you ignore the placement of the root, all five trees have the same topology, namely:

Note, the branch lengths are clearly different, these are all different trees, but the topology of the unrooted tree is the same in all of the above cases.

The following two trees are actually the same (within the limits of my drawing accuracy). The branch lenghts are the same, and the connectivity is the same. The fact that we rotate around one branch does not change the tree.

Would following tree also be the same?

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Yes, (except for minor branch lengths differences) this is the same tree, only rotated around a different branch.

Are the two trees below the same?

They have the same connectivity or tree topology, but the branch lengths are very different; i.e.,
Different trees, same tree topology

With four OTU (operational taxonomic units, often sequences, or species) there are only three possible unrooted tree topologies

But there is an infinite number of different trees, because the tree also can differ in branch lenghts, even if they have the same topology..

Are the next two trees the same? Same tree topology? Same unrooted tree topology?

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Different trees;

different tree topolgy, when considered as rooted trees;

same tree topology, when considered as unrooted.