  We adopt a maximum-likelihood framework to estimate parameters of a
stochastic susceptible-infected-recovered (SIR) model with contact tracing on a
rooted random tree. Given the number of detectees per index case, our estimator
allows to determine the degree distribution of the random tree as well as the
tracing probability. Since we do not discover all infectees via contact
tracing, this estimation is non-trivial. To keep things simple and stable, we
develop an approximation suited for realistic situations (contract tracing
probability small, or the probability for the detection of index cases small).
In this approximation, the only epidemiological parameter entering the
estimator is $R_0$.
  The estimator is tested in a simulation study and is furthermore applied to
covid-19 contact tracing data from India. The simulation study underlines the
efficiency of the method. For the empirical covid-19 data, we compare different
degree distributions and perform a sensitivity analysis. We find that
particularly a power-law and a negative binomial degree distribution fit the
data well and that the tracing probability is rather large. The sensitivity
analysis shows no strong dependency of the estimates on the reproduction
number. Finally, we discuss the relevance of our findings.
