Assuming that N is reported in the participants section and that all results are provied with p values and effect sizes (e.g. d): Which additional information can be acquired by test statistics (e.g. t values) and degrees of freedom? (see APA guidelines)
Short: they don't provide any practically relevant extra-information (beyond the specification of the model and the test, the sample size, and the p-value), but they allow to check if the test was performed correctly (regarding the statistical model and possible pseudo-replications).
The test statistic is rather technical and its meaning depends on the test. The t-value would tell you the ratio of the value of an estimate (e.g. a mean difference) and its standard error (wheras d is the ratio of a mean difference and the pooled standard deviation). The degrees of freedom (d.f.) tell you something about the sample size or the model complexity. It is the difference between the total sample size and the number of parameters in the statistical model. Given you know the model, you can easily calculate the sample size on which the test statistic and the p-value are based upon. For a two-sample t-test you know that the statistical model has two parameters (an intercept and a difference), and if 48 d.f. are reported, you know that 48+2 = 50 was the total sample size. If you know the sample size, you can infer the model complexity from the d.f. If you know that the sample size was 16 and d.f. = 11, you know that the model used estimated 5 parameters.
Knowing the d.f. and the sample size allows you to check if the reported statistics refer to the correct model (e.g. if the model included the covariables that should be adjusted for, or if metric predictors were modeled as categorical etc.). It also allows you to see if there might be a problem with pseudo-replication (if the d.f. are larger than the number of independent samples).
PS: I don't like relative effect sizes like d, because they blend two by itself important pieces of information: the absolute effect and the variance. If you don't have an idea about the practical relevance of some effect, you don't know what relative effect would be relevant either. A relative effect (d) can be "big" (e.g. according to Cohen, like d>1) for an irrelevantly small effect just because the variance is tiny (for whatever reason). A very relevant effect may be relatively small (like d < 0.1) just because a huge variance (for whatever reason).
They are required since often authors are in error (Nuijten, M. B., Hartgerink, C. H. J., van Assen, M. A. L. M., Epskamp, S., & Wicherts, J. M. (2016). The prevalence of statistical reporting errors in psychology (1985-2013). Behavior
Research Methods, 48, 1205-1226. DOI: 10.3758/s13428-015-0664-2 ) so this allows a way to check for this, and they don't take up much room in the paper.
Computed t compared with t from t- distribution table helps to make valid decisions and conclusions e.g. fit or coincidence of surfaces between geoid and mean sea level in vertical datum issues
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