Final answer:
In this binary tree implementation, the fraction of the total space taken up by overhead can be represented as 1/3.
Step-by-step explanation:
In this binary tree implementation, each internal node only stores two pointers and no data field, while the leaf nodes only store a data field. The pointers require 4 bytes each and the data objects require 6 bytes each. Let's consider the overhead space required for a full binary tree.
In a full binary tree, the number of leaf nodes is half the number of total nodes (n/2) and the number of internal nodes is half minus one (n/2 - 1) where n is the total number of nodes.
So, to calculate the overhead space, we need to determine the number of internal nodes and leaf nodes in the tree. Based on the given information, each internal node requires 8 bytes (2 pointers * 4 bytes) and each leaf node requires 6 bytes (1 data object).
Let's assume that the total number of internal nodes is i and the total number of leaf nodes is l. From the properties of a full binary tree, we know that the total number of nodes can be given by:
n = i + l
Since each internal node only stores pointers and no data field, the total space required for internal nodes is:
internal space = i * 8 bytes
Since each leaf node stores a data field, the total space required for leaf nodes is:
leaf space = l * 6 bytes
The total space required for the binary tree is:
total space = internal space + leaf space
Now, let's calculate the number of internal nodes and leaf nodes:
Since the tree is full, the total number of nodes can be given by:
n = i + l
Each internal node has two children, so we can also express the total number of nodes as:
n = 2 * i + 1
Simplifying the equation, we get:
i = (n - 1) / 2
Substituting the values of internal space and leaf space into the equation for the total space, we have:
total space = [(n - 1) / 2] * 8 + [n / 2] * 6
Finally, to find the fraction of the total space taken up by overhead, we divide the overhead space by the total space:
overhead fraction = [(n - 1) / 2] * 8 / ([(n - 1) / 2] * 8 + [n / 2] * 6)
In this case, the fraction will depend on the size of the binary tree (the total number of nodes). To simplify the fraction, we can consider the smallest possible binary tree, which has three nodes (i.e., one internal node and two leaf nodes). Substituting this value into the overhead fraction equation, we get:
overhead fraction = [(3 - 1) / 2] * 8 / ([(3 - 1) / 2] * 8 + [3 / 2] * 6) = 1/3
So, in the case of the smallest possible binary tree, the fraction of the total space taken up by overhead is 1/3.