Такая проблема: при удалении корневого элемента с обоими поддеревьями не выполняется балансировка. Исправь метод delet так, чтобы в этом случае выполнялась балансировка:
void balance1(Node* p, bool& h)
    {
        Node* p1, * p2;
        short b1, b2;
        if (p == NULL)
            return;
        switch (p->balance)
        {
        case -1:
            p->balance = 0;
            break;
        case 0:
            p->balance = 1;
            h = true;
            break;
        case 1: // балансировка
            p1 = p->right;
            b1 = p1->balance;
            if (b1 >= 0) // R-поворот
            {
                p->right = p1->left;
                p1->left = p;
                if (b1 == 0)
                {
                    p->balance = 1;
                    p1->balance = -1;
                    h = false;
                }
                else
                {
                    p->balance = 0;
                    p1->balance = 0;
                }
                p = p1;
            }
            else // RL-поворот
            {
                p2 = p1->left;
                b2 = p2->balance;
                p1->left = p2->right;
                p2->right = p1;
                p->right = p2->left;
                p2->left = p;
                if (b2 == 1)
                    p->balance = -1;
                else
                    p->balance = 0;
                if (b2 == -1)
                    p1->balance = 1;
                else
                    p1->balance = 0;
                p = p2;
                p2->balance = 0;
            }
        }
    }
    void balance2(Node* p, bool& h)
    {
        Node* p1, * p2;
        short b1, b2;
        switch (p->balance)
        {
        case 1:
            p->balance = 0;
            break;
        case 0:
            p->balance = -1;
            h = true;
            break;
        case -1:
            p1 = p->left;
            b1 = p1->balance;
            if (b1 <= 0) // L-поворот
            {
                p->left = p1->right;
                p1->right = p;
                if (b1 == 0)
                {
                    p->balance = -1;
                    p1->balance = 1;
                    h = false;
                }
                else
                {
                    p->balance = 0;
                    p1->balance = 0;
                }
                p = p1;
            }
            else // LR-поворот
            {
                p2 = p1->right;
                b2 = p2->balance;
                p1->right = p2->left;
                p2->left = p1;
                p->left = p2->right;
                p2->right = p;
                if (b2 == -1)
                    p->balance = 1;
                else
                    p->balance = 0;
                if (b2 == 1)
                    p1->balance = -1;
                else
                    p1->balance = 0;
                p = p2;
                p2->balance = 0;
            }
        }
    }

    void delet(std::string pn, int d, int m, int y, Node*& p, bool& h)
    {
        Node* q;
        if (p == NULL)
        {
            h = false;
        }
        else if (pn < p->key)
        {
            delet(pn, d, m, y, p->left, h);
            if (h)
                balance1(p, h);
        }
        else if (pn > p->key)
        {
            delet(pn, d, m, y, p->right, h);
            if (h)
                balance2(p, h);
        }
        else
        {
            if (p->left == NULL && p->right == NULL)
            {
                p->ht.removeItem(d, m, y);
                if (p->ht.isEmpty())
                {
                    delete p;
                    p = NULL;
                    h = true;
                }
            }
            else if (p->left != NULL && p->right == NULL)
            {
                p->ht.removeItem(d, m, y);
                if (p->ht.isEmpty())
                {
                    Node* temp = p->left;
                    delete p;
                    p = temp;
                    h = true;
                }
            }
            else if (p->left == NULL && p->right != NULL)
            {
                p->ht.removeItem(d, m, y);
                if (p->ht.isEmpty())
                {
                    Node* temp = p->right;
                    delete p;
                    p = temp;
                    h = true;
                }
            }
            else
            {
                p->ht.removeItem(d, m, y);
                if (p->ht.isEmpty())
                {
                    if (p->right == NULL) {
                        Node* temp = p->left;
                        delete p;
                        p = temp;
                        h = true;
                    }
                    else {
                        Node* temp = p->left;
                        Node* parent = p;
                        while (temp->right != NULL)
                        {
                            parent = temp;
                            temp = temp->right;
                        }
                        if (parent != p)
                            parent->right = temp->left;
                        else
                            parent->left = temp->left;
                        p->key = temp->key;
                        p->ht = temp->ht;
                        delete temp;
                        h = true;
                        
                    }
                }
            }
        }
        if (h && p != NULL)
        {
            if (p->balance == -2)
            {
                if (p->left->balance == 1)
                    balance2(p, h);
                else
                    balance1(p, h);
            }
            else if (p->balance == 2)
            {
                if (p->right->balance == -1)
                    balance1(p, h);
                else
                    balance2(p, h);
            }
        }
    }