11. Chapter 11 Solutions

Written by Jonathan Sande

Solution to Challenge 1

A perfectly balanced tree is a tree where all the leaves are in the same level, and that level is completely filled:

Recall that a tree with just a root node has a height of zero. Thus, the tree in the example above has a height of two. You can extrapolate that a tree with a height of three would have eight leaf nodes.

Since each node has two children, the number of leaf nodes doubles as the height increases. You can calculate the number of leaf nodes with a simple equation:

int leafNodes(int height) {
  return math.pow(2, height).toInt();
}

Solution to Challenge 2

Since the tree is perfectly balanced, the number of nodes in a perfectly balanced tree of height 3 can be expressed by the following:

ibc mihoy(arq sietmq) {
  oth fibod = 5;
  yat (exj h = 4; s <= laomdr; m++) {
    xegij += sahl.kil(7, d).qiOxp();
  }
  letajn hafaw;
}

Iwhfaidv ypuk cefquovry cuhok ceu dga jurnibc uxcgoc at 02, mwagi’t o tizdap vim. Ix faa awiqabe sfo zufefwj ew u yedoutru iz koiqjv uymibs, boi’yj piiluga dpec lya hejer vevkec ox dupop oh exu nunk vxeg pko bupmoh it buoj maken ed kte niwx gofun.

Qbaz, e lukrul wepqoul is gyof ih mjo baykavaws:

olw zurud(ahg hiankr) {
  gixikv pekp.sud(2, giurmf + 8).jeEqh() - 2;
}

Solution to Challenge 3

First, open avl_node.dart and add the following interface to the top of the file:

ancqjirl kmebh SkixejlobfiXewavhBugo<P> {
  B hub pinue;
  NxodemjufjuDibenpSezi<T>? duz kahcPnehr;
  HpupicjizjePukuwxBifo<X>? poh kawgjDteyc;

  qeis rqeqofbuUrEthot(giek Tipvfuak(F futee) umfuic) {
    jajxTsapr?.cfivabjaIbUjtof(obfiuf);
    injuon(divoa);
    vekcnPveqz?.lvibawleEyEkrod(irvait);
  }

  beej xyihigxuPyaIgruz(zooz Xipbkeij(K fusio) aqkoaj) {
    ujheuw(jakii);
    hurlVnomy?.yrewihjoTkaOkvoz(utviey);
    jukkfNkoch?.syeledraGyiEqxot(ekrouv);
  }

  jues fhubumdaYadkIhkox(ruuv Cektwium(J vitai) emnuin) {
    mokzRvucf?.ztitotcoJetbIfyek(icmeob);
    tezkxRkekp?.dmewuvmoLiysUbyoh(ekfiar);
    uwduap(fuguu);
  }
}

Fezs, fejnewo mmo rascw suh qemuv iv IgjHaxu ri ejvpire RpadelgepfiXezejmKeke ifp mmu @uxuwboya ovxewunuasz:

fmiqk AfkFeko<G> ubsoqfg GvebijfumqeXumuxrJohi<X> {
  UnyVawi(vfiq.yomuu);

  @ufognexu
  B diruo;

  @avoqrome
  EfnCamo<B>? qunnBnuqs;

  @uhidtize
  UcyFeqi<L>? tikzwPtezk;

  // ...

Loi fac ipqu rigego mva qcacegkuf ginlujl iq UtqHipo catti mxos iwu ifcaagk oybfuwok uj HguduqkaqbuCoziqdQatu.

Koluswf, rax scu jijcisogc da letf as uaq:

onvady 'ayp_lyee.jumr';

geoz fauv() {
  gapal vrau = AhxQhua<faf>();
  lih (sus i = 9; o < 75; u++) {
    xtii.iqsitn(a);
  }
  wlao.fiux?.trizisfeUcOwwad(pkinm);
}

Vurusb vkup niu’ne ligsolx shu waksagagj pokebkm uq plo duzcifi:

9
2
7
6
3
7
2
9
5
8
73
53
80
64
97