What are long-term trends in genealogy? While you could mean several things by the question, what about the word, and related words?
You may have seen reference to a new facility in Google Labs. Google Books Ngram Viewer "displays a graph showing how phrases have occurred in a corpus of books over the selected years. There's a more detailed explanation here.
Let's look at "genealogy", "family history" and "ancestry" from 1800 to 2000 in the English language set with a three year smoothing. "Genealogy" is the most frequently used until the second half of the 19th century when it is overtaken by "ancestry." "Genealogy" declines to about 1920 before making a slow recovery. "Ancestry" continues to become more popular until around 1920, declines but remains the most popular. "Family history" hardly shows at the start of the period, catches up with "genealogy" in the 1910s and continues to move with it. All three terms see an increase in the 1990s.
The British English corpus, a sub-set of the English set, shows a similar trend except that the use of "family history" skyrockets at the end of the period.
A second example is for the words, birth, marriage and death. There are many more occurrences of these words. "Death" has a sharp decline from 1850 to 1950 with still more than twice as many occurrences as "marriage: and "birth" which remain relatively steady through the two century period. Does this reflect a declining death rate and increasing longevity?
Finally, the words, "DNA", "database" and "Internet" found in modern genealogy all show the expected late 20th century increase.
Give it a try with words of your choice at http://ngrams.googlelabs.com