There are several layers to this onion.
First off, let me play devil's advocate and assume I am the analyst and took at quick look at these data. You have 3 time spans, 2002-2012 (45), 2013 (19), and 2014 (4). Converting to a yearly rate, the 2002-2012 time span had 4.09/year (45/11), so comparing that to the 4 recorded death penalties in 2014, I might conclude off the bat that save for 2013, the yearly rate hasn't changed.
But let's move on. There are a few ways to attack this problem, but in each you have some unknowns that need addressing. You suggest going the likelihood ratio route, in which case you need to specify the nature of the distributions, but I think this problem can be addressed with chi-square. In fact, your H0 and H1 as stated seems to anticipate this, as you are reducing everything to a contingency table (<=2012 vs > 2012). I will proceed with the assumption that the 2012 cut-off is adequate and call them the 'old' and 'new' time spans for convenience.
What you'd need to do next is figure out what you are comparing this death penalty to, and we'll have to make some assumptions. You might want to compare the rates for new and old versus population in country A. For present purposes, I will instead assume that exactly 20 people per year are convicted of a crime potentially punishable by death penalty, but receive a lesser penalty (say, life in prison or something).
Lets create a contingency table.
Note that I've converted the 45 for the old period to 4.09/year, then rounded down to 4 for convenience. For the new period, you actually have (19+4)/2 = 11.5, which I also round to 12 for convenience. Despite the simplification, the point should be clear.
Next, I assume that 20 people/year have death-penalty eligible convictions, which I add to the contingency table. You could use some other value too, but you'd have to correct the previous entries as well. E.g., rate per capita, rate per 1000 people, etc.
With this toy example set, we can then compute the chi-square, which I won't do by hand, but here is some sloppy R code that computes it.
x1 = data.frame(status=rep('Death',4),year='old')
x2 = data.frame(status=rep('Death',12),year='new')
x3 = data.frame(status=rep('No Death',20),year='old')
x4 = data.frame(status=rep('No Death',20),year='new')
x = rbind(x1,x2,x3,x4)
chisq.test(table(x),correct = FALSE)
The test does not attain conventional (p<.05) significance, suggesting that you retain the null hypothesis that there is no difference between the 'old' and 'new' periods. But, of course half the contingency table is fabricated.
Is your hypothesis reasonable? It might be, provided that you really don't have access to the yearly data, and are limited only to 2002-2012, 2013, 2014. If you do have access to the yearly data, simplifying like this doesn't make sense.