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16.6: Binary Searches

Last updated

Feb 19, 2022
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- 16.5: Spanning Trees
- 16.7: Activities

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Definition: Binary Search Tree

a tree in which every node has degree $1$ or $3\text{,}$ except for a single node of degree $2$

Definition: Initial Node

the unique node of degree $2$ in a binary search tree

Definition: Terminal Node

a node of degree $1$ in a binary search tree

Definition: Binary Search

the construction of a binary search tree through a series of “either-or” decisions

Example $\PageIndex{1}$

Estimate the root of $f(x) = 4x^3 + 6x^2 + 3x - 1$ that lies in $(0,1)$ to $2$ decimal places.

Solution

The Intermediate Value Theorem from first-year calculus says that if $f$ is continuous on the closed interval $[a,b]$ and $f(a),f(b)$ are nonzero and opposite signs, then $f$ has a root in the open interval $(a,b)\text{.}$ We have $f(0) = -1 \lt 0$ and $f(1) = 12 \gt 0\text{,}$ so there is indeed a root in $(0,1)\text{.}$ The graph in Figure $\PageIndex{1}$ was obtained by performing a binary search by splitting into subintervals.

Figure $\PageIndex{1}$ : A binary search tree search for the root of a polynomial.

Since $f(0.225) \gt 0\text{,}$ the root must be in the subinterval $(0.22,0.225)\text{.}$ This tells us to round down to $0.22$ instead of rounding up to $0.23\text{,}$ so we conclude that the root is approximately $0.22\text{.}$

Definition: Binary Search Tree

Definition: Initial Node

Definition: Terminal Node

Definition: Binary Search

Example 16.6.1\PageIndex{1}

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Example $\PageIndex{1}$