is a deterministic algorithm for breadth-first traversal possible (binary heap)
Binary trees can have different structures. Every node can have at most two children. At most means you might have 2 children, you might have one child or no child. This leads to different structural possibilities for the binary tree.

There is no definite pattern to tell how to find a particular node. Closest you can get to a pattern is when you have a binary heap, but in a heap it is fix that the binary tree will be complete binary tree i.e. all the internal nodes will have two children and the leaf nodes at the last level will be as left as possible. For a heap you can determine the left child as $2*i + 1$ and right child as $2*i+2$ where $i$ is the index of node and $i = 0$ for root node.

In the worst case a binary tree can degenerate to a linked list which means you would have to proceed linearly to find the element you want. So, we can say how will the search of a node proceed totally depends on the structure of the binary tree.

To find a node basically traverse the tree with any of DFS or BFS.