Steve Sailer posts the following map:
It is explained thusly by New York Times blogger Frank Jacobs:
* The “absolute nuclear” family type is both liberal and unequal. Children are totally emancipated, forming independent families of their own. The inheritance usually goes to one child, often a son.
* The “egalitarian nuclear” family type is both liberal and equal. Children are as emancipated and independent as in the previous type, but equal division of the inheritance encourages stronger parent-children relations before the passing of the parents.
* The “stem” family type is both authoritarian and unequal. Several generations live under one roof, with one child marrying to continue the line. The other children remain unmarried at home, or leave to get married.
* The “incomplete stem” family is as above, but with slightly more equal inheritance rules — an intermediate with the last family type.
* The “communitarian” family is both authoritarian and equal. All sons can marry and bring their wives into the ancestral home. The family inheritance is divided equitably among all children.
Is there a connection between family-inheritance type and fertility in today’s Europe? Have a look:
[Map from Eurostat]
How does “family type” affect fertility?
I have a hard time seeing seeing any strong correlation. More precise data would help. Certainly other factors seem more important, political, cultural, economic. German fertility is low across the board, across two family-type zones. Nevertheless, other ‘Stem Family’ regions have more robust fertility. Germans are clearly still very nationally — perhaps better-to-say ‘civilizationally’ — pessimistic, a legacy of defeat decades ago and demonization ever since.
A more suitable question would be, “all else equal, how does ‘family-type’ affect fertility?” Inheritance is a major part of this ‘family-type’ concept, and the question of inheritance and fertility was first brought to my attention by the commenter Rollory in a post about France’s low-fertility 1800s.
A casual analysis: Looking at Netherlands alone (all are part of Dutch culture), it seems Absolute Nuclear regions out-breed Stem Family regions. The highest-fertility department in France seems to be mostly Absolute Nuclear. The highest-fertility region of Scotland is Absolute Nuclear.
Why would “Absolute Nuclear” produce highest fertility?
Hail To You 
in “Europe”, but maps cover only less than half of Europe…
I studied Emmanuel Todd’s work a bit for an academic project. (That top map is surely drawn from his 1990 ‘The Invention of Europe’, based on Frédéric LePlay’s work from the 1850s.) He has in fact mapped those family structures to the planetary level. You can see the family structure data extended to Eastern Europe here.
Demographer Giuseppe Micheli has asked a similar question to yours, but only for southern Europe (Kinship, Family and Social Network: The anthropological embedment of fertility change in Southern Europe). That’s quite a long and involved paper, and his major question is, why has fertility plummeted in strong-kinship Mediterranean Europe? Is there a possible link between that and their old family types?
He doesn’t really come up with convincing answers, and I haven’t either. Traditionally, regions inside the Hajnal Line have had later marriage and lower TFR, but looking at the modern Eurostat map you posted, that seems to have partially reversed itself.
Speaking for France, where I live, they’ve enforced strong pro-natalist policies (generous maternity leave w/ guaranteed job when you come back, state-funded day care, ‘child payments’ to any woman who has a kid [not just poor women], etc.) This may help account for their high TFR.
(Re: French fertility crash in the 19th c., I added a comment to that thread from a French-language source I found and yes, among others they cite the equal-inheritance law as a contributing factor.)
I believe it’s due to smaller habitable area (much of Spain is desert, Italy is extremely narrow, and much of Greece is given to tourist developments) and runaway urbanization — the cities of these nations have exerted a draw upon the surrounding populations much stronger than northern nations, I suppose in the rush to escape their notorious backwardness. Europeans in crowded conditions always respond by restricting natality, whereas Central Americans (‘Latinos’) have responded to similar conditions by reproducing more, probably owing to their evolutionary dependence on corn as a staple food and low level of individualism. Pretty straightforward r/K perspective.