A kingdom consists of a king, his children, his grandchildren, and so on. Every once in a while, someone in the family dies or a child is born.

The kingdom has a well-defined order of inheritance that consists of the king as the first member. Let's define the recursive function `Successor(x, curOrder)`, which given a person `x` and the inheritance order so far, returns who should be the next person after `x` in the order of inheritance.

    Successor(x, curOrder):
        if x has no children or all of x's children are in curOrder:
            if x is the king return null
            else return Successor(x's parent, curOrder)
        else return x's oldest child who's not in curOrder

For example, assume we have a kingdom that consists of the king, his children Alice and Bob (Alice is older than Bob), and finally Alice's son Jack.

1. In the beginning, `curOrder` will be `["king"]`.
2. Calling `Successor(king, curOrder)` will return Alice, so we append to `curOrder` to get `["king", "Alice"]`.
3. Calling `Successor(Alice, curOrder)` will return Jack, so we append to `curOrder` to get `["king", "Alice", "Jack"]`.
4. Calling `Successor(Jack, curOrder)` will return Bob, so we append to `curOrder` to get `["king", "Alice", "Jack", "Bob"]`.
5. Calling `Successor(Bob, curOrder)` will return `null`. Thus the order of inheritance will be `["king", "Alice", "Jack", "Bob"]`.

Using the above function, we can always obtain a unique order of inheritance.

Implement the `ThroneInheritance` class:

- `ThroneInheritance(string kingName)` Initializes an object of the `ThroneInheritance` class. The name of the king is given as part of the constructor.
- `void birth(string parentName, string childName)` Indicates that `parentName` gave birth to `childName`.
- `void death(string name)` Indicates the death of `name`. The death of the person doesn't affect the `Successor` function nor the current inheritance order. You can treat it as just marking the person as dead.
- `string[] getInheritanceOrder()` Returns a list representing the current order of inheritance excluding dead people.

Example 1:

    Input
    ["ThroneInheritance", "birth", "birth", "birth", "birth", "birth", "birth", "getInheritanceOrder", "death", "getInheritanceOrder"]
    [["king"], ["king", "andy"], ["king", "bob"], ["king", "catherine"], ["andy", "matthew"], ["bob", "alex"], ["bob", "asha"], [null], ["bob"], [null]]
    Output
    [null, null, null, null, null, null, null, ["king", "andy", "matthew", "bob", "alex", "asha", "catherine"], null, ["king", "andy", "matthew", "alex", "asha", "catherine"]]

    Explanation
    ThroneInheritance t= new ThroneInheritance("king"); // order: king
    t.birth("king", "andy"); // order: king > andy
    t.birth("king", "bob"); // order: king > andy > bob
    t.birth("king", "catherine"); // order: king > andy > bob > catherine
    t.birth("andy", "matthew"); // order: king > andy > matthew > bob > catherine
    t.birth("bob", "alex"); // order: king > andy > matthew > bob > alex > catherine
    t.birth("bob", "asha"); // order: king > andy > matthew > bob > alex > asha > catherine
    t.getInheritanceOrder(); // return ["king", "andy", "matthew", "bob", "alex", "asha", "catherine"]
    t.death("bob"); // order: king > andy > matthew > bob > alex > asha > catherine
    t.getInheritanceOrder(); // return ["king", "andy", "matthew", "alex", "asha", "catherine"]

Constraints:

- `1 <= kingName.length, parentName.length, childName.length, name.length <= 15`
- `kingName`, `parentName`, `childName`, and `name` consist of lowercase English letters only.
- All arguments `childName` and `kingName` are distinct.
- All `name` arguments of `death` will be passed to either the constructor or as `childName` to `birth` first.
- For each call to `birth(parentName, childName)`, it is guaranteed that `parentName` is alive.
- At most `105` calls will be made to `birth` and `death`.
- At most `10` calls will be made to `getInheritanceOrder`.