diff --git a/docs/Chap16/16.3.md b/docs/Chap16/16.3.md index 0d3288916d..3e57a875ec 100755 --- a/docs/Chap16/16.3.md +++ b/docs/Chap16/16.3.md @@ -50,7 +50,7 @@ B(T) & = f(x) d_T(x) + f(y) d_T(y) \\\\ \end{aligned} $$ -Thus, the statement of theorem is true. Now suppose $n > 2$ and also suppose that theorem is true for trees on $n - 1$ leaves. +Thus, the statement of theorem is true. Now suppose $n > 2$ and also suppose that theorem is true for trees on $n - 1$ leaves. Let $c_1$ and $c_2$ are two sibling leaves in $T$ such that they have the same parent $p$. Letting $T'$ be the tree obtained by deleting $c_1$ and $c_2$, by induction we know that $$ @@ -88,7 +88,7 @@ $$ \end{aligned} $$ -That is, where $i$ and $j$ are the upper and lower median, +That is, where $i$ and $j$ are the upper and lower median, respectively; and $c_i = d_T(x_i) - d_T(x_{n - i + 1})$: $$f(x_1)c_1 + \cdots + f(x_i)c_i > f(x_j)c_i + \cdots + f(x_n)c_1$$ @@ -108,7 +108,7 @@ $$ First observe that any full binary tree has exactly $2n - 1$ nodes. We can encode the structure of our full binary tree by performing a preorder traversal of $T$. For each node that we record in the traversal, write a $0$ if it is an internal node and a $1$ if it is a leaf node. Since we know the tree to be full, this uniquely determines its structure. -Next, note that we can encode any character of $C$ in $\lceil \lg n \rceil$ bits. Since there are $n$ characters, we can encode them in order of appearance in our preorder traversal using ndlg ne bits. +Next, note that we can encode any character of $C$ in $\lceil \lg n \rceil$ bits. Since there are $n$ characters, we can encode them in order of appearance in our preorder traversal using $n\left\lceil \lg n \right\rceil$ bits. ## 16.3-7