How do binary trees use memory to store its data?
We use implicit array representation to store Binary Heaps, where if a node has an index $i$, its children are found at indices $2i + 1$ (for the left child) and $2i +2$ (for the right), assuming indices start at 0.

Now this is not wasteful for Binary Heaps since nodes are added in Breadth First method. However for normal Binary Trees implicit array representation is very expensive to grow and wastes space in the order of $O(2^h)$ where $h$ is the height of the tree (Consider skewed binary trees).

For trees having a large number of nodes we use Succint Data Structures as pointed out by Raphael. A normal binary tree of $n$ nodes is stored using $O(nlogn)$ bits (pointer representation), but succinct representations require just $2n + o(n)$ bits and carry out many sophisticated operations in constant time. This link gives us a good overview of a simple representation of succint representations.