## The law of the constancy of the number of ancestors

The number of ancestors in each generation, in any branch of a tree, strives for a constant value (range).

In practice, it is advisable to consider branches with constant conditions (locality, estate, religion, nationality, etc.).

The proposed law is closely related to the three previous ones, one of which, the law of decreasing ancestors begins to work when a pair of ancestors having common parents or one common parent meet in the ascending tree. In other words, when brothers and / or sisters (siblings or stepbrothers) appear in the ascending tree.

Suppose that in the generation n+1, the first reduction of the ancestors occurred. This was due to the fact that in generation n (in which the number of ancestors is 2^{n}, if we consider ourselves in the zero generation), brothers and / or sisters appeared who may be in the same generation of n or earlier.

Given that brothers and / or sisters are children of the same parents, the number of generations before them will be approximately the same. But since the age of the brothers and / or sisters can differ by 20 years (about 1 generation), and there is also a significant probability that the activity of 3 or 4 generations can fall out for a hundred years, it is most likely that the brothers and / or sisters will be in one generation, neighboring or through one generation.

Thus, if we consider the possibility of reduction at the expense of brothers and / or sisters for three generations, then 3 options are possible.

Each brother and / or sister must have two ancestors one generation above them. Moreover, the decrease in ancestors occurs at the expense of parents who are in the older generation. In these examples, the reduction occurs in generation n+1.

Denote by L_{m}, L_{m-1}, L_{m-2} – the number of ancestors in m, m-1 and m-2 generations respectively. For the first generation in which ancestor reduction (n+1) occurred, these values are respectively 2^{n}, 2^{n-1}, 2^{n-2}.

Then, in the first case, the number of possible combinations of pairs of brothers and / or sisters (hereinafter VKPBiS) is equal.

In fact, since a marriage between a brother and sister is unlikely, then this value could be considered equal to (L_{m-1})(L_{m-2})/2, but we neglect this reduction or believe that this is possible.

In the second – L_{m}xL_{m-1}. At the same time, the number of ancestors (L_{m}) should be reduced by 2, since usually parents cannot be brothers and sisters of their children. But we neglect this reduction or believe that this is possible. That is, an ancestor from generation m can be a brother or sister to any ancestor from generation m-1.

In the third case, L_{m}xL_{m-2}. That is, an ancestor from generation m can be a brother or sister to any ancestor from generation m-2.

Thus, the total number of VKPBiS for generation n+1, taking into account their presence in three generations, is: L_{n}^{2}/2+L_{n}xL_{n-1}+L_{n}xL_{n-2}, and as a result of these VKPBiS in the next generation ancestors were reduced.

Suppose that, on average, a decrease of 1 ancestor results from K VKPBiS. The coefficient K is called the reduction coefficient. At the same time, these VKPBiS include a set of differently probable events (brother and / or sister of the spouse / not spouse, from the same / different generations, one / two common ancestors, etc.), as a result of which one intersection of the ancestors occurs.

At the same time, 2^{n-1}(2^{n-1}/2+2^{n-2}+2^{n-3})<K≤2^{n}(2^{n}/2+2^{n-1}+2^{n-2}) or consequently 1.25×2^{2(n-1)}<K≤1.25×2^{2n}, where n is the last generation without ancestral reduction. If K is outside the specified range, then n must change its value.

**Example 1**

Since K is a certain average value for specific conditions, it can be applied to any subsequent generation in which a proportional decrease in ancestors will occur.

Then, in the generation m+1(m+1>n), the number of VKPBiS will be: L_{m}^{2}/2+L_{m}xL_{m‑1}+L_{m}xL_{m-2}, and accordingly, the double value of the number of ancestors in the previous generation will decrease (2L_{m}) on (L_{m}^{2}/2+L_{m}xL_{m-1}+L_{m}xL_{m-2})/K ancestors. As a result, in the generation m+1, the number of ancestors will be equal to:

2L_{m}-(L_{m}^{2}/2+L_{m}xL_{m‑1}+L_{m}xL_{m-2})/K=L_{m}(2-(L_{m}/2+L_{m-1}+L_{m-2})/K) (1)

**Example 2**

It can be seen from formula (1) that if (L_{m}/2+L_{m-1}+L_{m-2})<K, then the number of ancestors in the next generation is larger than the previous one, if (L_{m}/2+L_{m-1}+L_{m-2})>K, then the number of ancestors in the next generation is less than the previous one, if (L_{m}/2+L_{m‑1}+L_{m-2})=K, then the number of ancestors remains unchanged.

It is clear that in the first generations after the reduction, an increase in the number of ancestors will be observed.

Let us find in general the number of ancestors (for the maximum reduction coefficient K=1.25×2^{2n} for the first reduction in the generation n+1:

*Table 1*

Generation | n-2 | n-1 | n | n+1 | |

Number of ancestors | 2^{n-2} | 2^{n-1} | 2^{n} | 2^{n+1}-1 | … |

Let us find in general the number of ancestors (for the maximum reduction coefficient K=1.25×2^{2(n+1)}=1.25×2^{2n}x4) for the first reduction in the generation n+2:

*Table 2*

Generation | n-1 | n | n+1 | n+2 | |

Number of ancestors | 2^{n-1} | 2^{n } | 2^{n+1} | 2^{n+2}-1 | … |

It can be seen from the above information that the reduction coefficient increased by 4 times, but the number of ancestors increased by 4 in generations n and n+1 (Table 2) in relation to generations n-2 and n-1 (Table 1), respectively, and in generation n+2 increased 4 times without one ancestor.

Thus, from the formula: L_{n}(2-(L_{n}/2+L_{n-1}+L_{n-2})/K) it can be seen that with an increase in the number of ancestors in each generation and a reduction factor of 4 times, the value of the number of ancestors in each generation also increases almost 4 times. And therefore, at different reduction coefficients, the form of the function remains unchanged, therefore, having studied the function for specific n and K, the results can be extrapolated to other values of n and K.

The values of the number of ancestors are almost identical with a factor of 4 and a difference of 2 generations.

You can see (Table 1,2) that when the reduction coefficient is increased by 2 times, the number of ancestors also increases by 2 times, starting from the next generation.

Moreover, all values of the number of ancestors with intermediate values of the reduction coefficients will be between the corresponding values of the number of ancestors.

Given the great uncertainty in practical genealogy, the reduction coefficient can be rounded to the nearest value equal to 1.25×2^{n}.

Since the function tends to a constant value, the value of the function L_{n}(2-(L_{n}/2+L_{n-1}+L_{n-2})/K)=C, but the function values for previous generations are equal to C(L_{n}=L_{n-1}=L_{n-2}=C). Then C(2-(C/2+C+C)/K=C, whence 2-2.5C/K=1, therefore, C=0.4K. That is, the number of ancestors strive for a constant value of 0.4K.

**Example 3**

In the case of a change in the initial data (input of weight coefficients in formula (1), a decrease or increase in the number of generations affecting the reduction coefficient), the number of ancestors can tend to either a constant value (with a different kind of function) or be in a certain range of infinitely alternating values. But in this study, the most probable initial data are used, on the basis of which the following can be stated.

The number of ancestors in each generation in any branch of the tree tends to a constant value equal to 0.4K (where K is the reduction coefficient equal to the number of possible combinations of pairs of brothers and / or sisters, which result in reduction by one ancestor). The maximum value (0.63K) is achieved in the 2n±1 generation, the minimum (0.21K) after 3 generations, where n is the last generation without ancestor reduction.

Good luck in finding.