is a deterministic algorithm for breadth-first traversal possible (binary heap) Imagine I have a binary heap, traversed in a breadth first manner like so:

                  0
       1          |           2
  3    |    4     |    5      |     6
7 | 8  |  9 | 10  |  11 | 12  |  13 | 14


To the left of root (0) is 1, to the right is 2, to the left of 1 is 3 to it's right is 4 etc.

So node 13 corresponds to the following pattern (right right, left, (or 0,0,1, perhaps)). 

The question is, is there an equation, function, or algorithm that will produce the pattern to get to any node without actually producing the entire tree and traversing it?

Say I didn't create the tree in memory (it's just a virtual tree). How would I know that in order to get to 8 I need to start at root and go: left,left, right (1,1,0)?

Is this computable without creating the tree, and traversing it? If so what is the pattern for 42? is it easily calculable (is it linear) ?

I'm sorry for not knowing all the CS terms (I'm not a cs major, but I want to know if there's a deterministic algorithm to produce a path to any node, if so, whats it's computational efficiency) ?
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.