CMSC 420: Data Structures
Spring 2000
MID TERM EXAM
Problem 1. (6 points) Consider a binary tree
T
whose nodes are labeled with alphabetic charac-
ters. Suppose you are told that
e; a; i; j; d; b;f;g;h;c
is an inorder traversal of this tree, and that
a; j; i; e; f; h; g; b; c; d
isa postordertraversalof thistree. Doesthis informationallow youto uniquely
reconstruct a binary tree? If yes, draw such a tree. If not, explain why either no such tree exists,or
why more than one such tree exists.
Problem 2. (8 points) Let
T
1
be a binary tree, and let
T
2
be a version of
T
1
which is threaded in pre-
order. Assume the existence of fields LTAG and RTAG (with “t” and “l” as possible values denoting
thread/link) respectively which are used to check if the left/right child is a linkor a thread. Suppose
you are given a pointer
P
to a node in tree
T
2
. Is it possible to find the preorder successor of the node
P
is pointing to? If this is possible, give an algorithm in pseudo-code. Otherwise explain why it is
not possible.
Problem 3. (16 points) You are familiar with quadtrees and
2
-d trees to represent
2
-dimensional
data. In this problem, we define a new data structure to represent
2
-d data called a ternary-tree. As in
the case of quadtrees and
2
-d trees, the root of a ternary tree represents a rectangular region
R
.Every
node
N
has fields
N:x; N:y
representing a point in
2
-space, fields
N:lef t; N :rig ht:N :top; N:bot
denoting the extremities of the rectangle (implicitly)represented by node
N
(and in which the point
N:x; N:y
lies), and three link fields N.NW,N.NE,N.SO denotingthe regions obtained by splitting
the rectangle sp ecified by
N:lef t; N :rig ht:N :top; N:bot
into three parts— the northwest, northeast,
and southern regions obtainedby first drawing a horizontal line through
N:x; N:y
to obtain two parts
— the southern part (pointed to by N.SO and the northern part. The northern part is then split in two
by drawing a vertical line through
N:x; N:y
to obtain the regions associated with the N.NW,N.NE
children of
N
. Assume all regions are closed on the left and bottom, and open on the top and right.
1. (2points) Suppose youknow allthe fieldsof a node
N
. Howwould youcompute
M:lef t; M :top
from the fields of
N
when
M
is the SO child on
N
.
2. (4 points) Construct the tree obtained when you insert the following points in the order shown:
(50,50), (25,30), (45,75),(20,35). Show all fields of all nodes.
3. (10 points) Write, in pseudocode, an algorithm that takes as input, the root
T
of a ternary tree,
and a point
(
a; b
)
, and returns the “furthest neighbor” of
(
a; b
)
in
T
, i.e. it returns that point
in
T
which is furthest away from
(
a; b
)
. When writing this algorithm, you may assume the
existence of functions called
mindist; maxdist
which take three arguments - an
x
coordi-
nate, a
y
-coordinate, and a pointer to a node
N
in the ternary tree.
mindist
(
a; b; N
)
returns
min
f
d
((
a; b
)
;
(
x; y
))
j
(
x:y
)
isin therectangularregion associatedwith
N
g
while
maxdist
(
a; b; N
)
returns
max
f
d
((
a; b
)
;
(
x; y
))
j
(
x:y
)
isin therectangularregion associatedwith
N
g
.
d
(
a; b
)
;
(
x; y
))
here is the distance between
(
x; y
)
and
(
a; b
)
.
1