Is this a philosophical question?

Viewed from the root  a tree is a tree. After traversing to the first level of nodes (viewed from the root), there will be n subtrees with the root being their predecessor. This process continues until you find trivial subtrees (subtree = leaf).

Naturally, you can only traverse a tree along the branches and it directly follows that you can only reach a node if you already reached its predecessor or successor. There is no difference if  you would use breadth-first or depth-first search for your problem.

Now, near is a very subjective thing. a branch (if a length is attached) can be of arbitrary length compared to any other branch in the same tree which may completely invalidate your  statement of nearness. Reaching something in one step in a tree traversal algorithm does not imply closeness. Some branches can be very long (longer than comparable subtreees attached to other nodes).

In this light, there is no guarantee for a predecessor or successor to be near to each other.

BTW, there is always a subtree in a tree with the trivial subtrees being the root and leaves.

There are different algorithms to traverse a tree. You have mentioned in-order traversal yourself. This belongs to the class of depth-first traversals. Other depth-first algorithms are pre-order and post-order. They will provide a entirely different picture of the tree traversal. Then, there is also breadth-first traversal. This means that you first walk through all nodes on the same level before going to the next level. Here you could actually put in some randomness and first shuffle the nodes of the same level before using them. Other algorithms do not make much sense because it is hard to cover the entire tree when not using one of these algorithms.

So, basically I cannot agree with your statement that predecessors and successors are close to each other. This totally depends on the algorithms. Then, the remaining question is a philosophical one: Why does the in-order algorithm produce in-order results?

We can not generally say that predecessors and successors are nearer to each other. It depends upon the traversal algorithm (type of tree walk algorithm) you are using. If its an in-order traversal with a binary search tree then they will be closer to each other. But as Simon mentioned, if its some other algorithm like DFS or BFS then the scenario is completely different. So the conclusion can be, we can not comment on successors and predecessors as the definition for them changes from one algorithm to another.

Hope it helps :)